The nodal set of solutions to sublinear equations

Nicola Soave, Politecnico di Milano
Tuesday, February 20, 2018 - 1:30pm to 3:30pm
PDL C-401

We are concerned with the nodal set of solutions to sublinear equations of the form
\begin{equation*}
-\Delta u = \lambda_+ \left(u^+\right)^{q-1} - \lambda_- \left(u^-\right)^{q-1} \quad \text{in $B_1$}
\end{equation*}
where \$\lambda_+,\lambda_- > 0\$, \$q \in [1,2)\$, \$B_1=B_1(0)\$ is the unit ball in \$\mathbb{R}^N\$, \$N \ge 2\$, and \$u^+:= \max\{u,0\}\$, \$u^-:= \max\{-u,0\}\$ are the positive and the negative part of \$u\$, respectively. This class includes equations with sublinear (\$1<q<2\$) and discontinuous (\$q=1\$) nonlinearity. In both cases the right hand side is not locally Lipschitz continuous as function of \$u\$.

In this talk we present results regarding:
(a) the validity of the unique continuation principle;
(b) the finiteness of the vanishing order at every point and the complete characterization of the order spectrum; (c) a weak non-degeneracy property; (d) the partial regularity of the nodal set of any solution: the nodal set is a locally finite collection of regular codimension one manifolds up to a residual singular set having Hausdorff dimension at most \$N-2\$ (locally finite when \$N=2\$) and a partial stratification theorem.

Ultimately, the main features of the nodal set are strictly related with those of the solutions to linear (or superlinear) equations, with two remarkable differences. First of all, the admissible vanishing orders can not exceed the critical value \$2/(2-q)\$. At threshold, we find a multiplicity of homogeneous solutions, yielding the non-validity of any estimate of the \$(N-1)\$-dimensional measure of the nodal set of a solution in terms of the vanishing order.

The proofs are based on Weiss-type monotonicity formulae, blow-up arguments and the classification of homogenous solutions.

The talk is based on joint works with Tobias Weth and Susanna Terracini.

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